Origin of Bellerophon states in globally coupled phase oscillators

We unveil the basic mechanisms and general conditions for the emergence of Bellerophon states, which are higher order coherent states appearing in globally coupled phase oscillators. The critical points for the involved phase transitions are determined analytically. The significant feature of Bellerophon states is that the oscillators' effective frequencies are locked to quantized plateaus, a point which is fully clarified on the basis of circle map theory. Each quantized plateau corresponds to a harmonic frequency of the Fourier decomposition of the order parameter. Our approach exploits the fact that the order parameter is always real, due to a special symmetry of the system which furthermore prevents the formation of even integer multiple plateaus of effective frequencies.